Vibration Analysis of Circular Cylindrical Shells - A Comparative study
A review of mathematical analysis on the natural frequencies of cylindrical shells reported by various researchers and comparison with those calculated using DMV theory is presented in this paper. The mathematical analysis by DMV and other theories are centred around frequency bands of 100 Hz to 5000 Hz which are most important and relevant for vibration damping and reduction of radiated noise. Cylindrical shells of lengths (l) 300 mm to 1500 mm, and radius (R) to thickness (h) ratio of 50-100 are considered for determination of natural frequencies by DMV theory. Predicted natural frequencies of various modes are found to be sharply decreasing beyond a certain l/R ratio, while increases with thickness as the length and radius are kept constant. The natural frequency decreases with increase in circumferential mode (n) up to a certain mode number, and then again increases. Comparison is made of DMV and other theories for various shells of a wide range of dimensions (l/R =1 to 10, R/h =2.5 to 100) in axial (m) and circumferential modes (n). Both aluminium and steel cylinders are considered because of their wide application in industrial piping. Their specific modulus (E/r ≈ 0.0255 GN.m/kg) are identical and hence their natural frequencies would be similar if all dimensions are identical. For aluminium cylinders with, R/h > 10, the DMV and other theories predict almost identical frequencies except nominal variations upto 17% in very few modes. For steel cylinders, at lower thicknesses (R/h=10-50), the differences are not more than 12%, while for higher thicknesses (R/h= 2.5 to 10), some theories predicted 20-28% difference than those by DMV theory. An experimental result on shells having R/h =19 and l/R= 8.3 covering natural frequency range of 1100 Hz to 13000 Hz showed very good match with DMV theory with 8% to 11.8% difference only for few cases, especially for n=2. It can be concluded that with at the most 12% difference, resonances in hollow circular cylindrical pipes of reasonable dimensions can be predicted by simple DMV theory for practical purpose, such as passive damping treatment.